As KConrad points out, you perhaps mean to say that $F$ is a finite extension of $\mathbf{Q}_2$ or of $\mathbf{F}_2((x))$, and that the quadratic extesnions $E|F$ is separable (and hence galoisian) in the second case.
With this interpretation of the question, $N_{E|F}(E^\times)$ is a closed subgroup of index $2$ in $F^\times$, and every closed subgroup of index $2$ in $F^\times$ is of this form. In particular, the ramification index $e_{E|F}$ does not determine the subgroup in question.
For more on this, see the relevant chapter in Serre's Corps locaux (=Local fields) or the book by Fesenko and Vostokov, among many other places.